One of the biggest winners in the sweeping Democratic victories in Senate races this month was a woman whose name appeared on no ballot: Sen. Patty Murray (D-Wash.). With a low-key style that contrasts with some of the Senate’s camera hogs, Murray may be the most powerful senator a whole lot of people have never heard of outside of the two Washingtons where she lives and works.

As chair of her party’s Senate campaign arm, the architect of surprising Democratic gains and the incoming chair of the powerful Senate Budget Committee, Murray now occupies a place of special influence in the Senate.

And so what Murray has to say about the “fiscal cliff,” a combination of tax increases and spending cuts set to take effect in January, may be of particular importance. In a town consumed by talk of the apocalyptic consequences of failing to resolve the budgeting crisis, Murray has been arguing that missing the deadline for a deal — going over the cliff — could actually make getting a deal easier.

“She’s low-key but very focused and very forceful,” said Sen. Claire McCaskill (D-Mo.), whose defeat was widely predicted but who won reelection in November after her Republican opponent claimed that pregnancy rarely results from rape. “If you know Patty and you work with her, you’d be a fool to underestimate her.”

Murray, 62, who holds the Senate’s No. 4 position, has repeatedly been handed jobs no one else wanted by Majority Leader Harry M. Reid (D-Nev.). Two years ago, she agreed to chair the Democrats’ 2012 senatorial campaign effort. At the time, it was widely assumed the job would mean presiding over the party’s loss of its majority. Republicans needed only four seats to prevail, and Democrats were defending 23 seats, compared with 10 for the GOP. Key retirements in several Republican-leaning states seemed to portend a GOP takeover.

Instead, through a combination of good candidate recruitment, good luck and Republican missteps, Democrats won almost every tough contest, expanded their effective majority by two seats and helped achieve several historic firsts, including a record number of women elected to the Senate. In a chamber of big egos, Murray’s success at achieving what had been thought impossible has given her new leverage with her Senate colleagues, who are especially grateful that no Democratic incumbent lost.

Now, Murray is using that influence to argue Democrats should not forget the tactical advantage they could gain in January, after the deadline for the fiscal cliff has passed. Starting with a speech at the Brookings Institution in July and continuing in a series of interviews last week, Murray, in typically non-bombastic fashion, has argued that Democrats shouldn’t take a bad deal in December when their political leverage will only increase in the new year.

That’s because next month, tax cuts first enacted under President George W. Bush will expire for everybody. Murray reasons that might make it easier to get Republicans to agree to reinstate the cuts only for the middle class and let the nation’s wealthiest 2 percent pay more toward the reducing the debt, as Democrats desire.

Suddenly, Congress would be voting to lower taxes for the vast majority of Americans, rather than agreeing to raise them for a few. “I’m hopeful we can get there. I’m an optimist” about a deal to avoid the cliff, she said in an interview. But, she added, “the dynamic changes dramatically on January 1st. . . . Anything we do will be a tax cut at that point, because taxes will have gone up.”

In a Congress of hot tempers and sharp tongues, Murray doesn’t favor over-the-top rhetoric. Once dismissed by a Washington state representative as just a “mom in tennis shoes,” she’s turned the moniker into a campaign symbol of determined strength.

Murray, colleagues agree, doesn’t issue idle threats.

“Everyone takes Senator Murray seriously because she does not bluster,” Reid said. “She simply says what she means and stands by it.”